It was invented by Wendy Carlos and used on three pieces on her album Beauty in the Beast (1986): Just Imaginings, That's Just It, and Yusae-Aisae. Versions of the scale have also been used by Ezra Sims and Frans Richter Herf.[3]

1.
Harmonic series (music)
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A harmonic series is the sequence of sounds where the base frequency of each sound is an integer multiple of the lowest base frequency. Pitched musical instruments are based on an approximate harmonic oscillator such as a string or a column of air. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling each other to form standing waves, interaction with the surrounding air causes audible sound waves, which travel away from the instrument. Because of the spacing of the resonances, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest frequency. The musical timbre of a tone from such an instrument is determined by the relative strengths of each harmonic. A complex tone can be described as a combination of many simple periodic waves or partials, each with its own frequency of vibration, amplitude, a partial is any of the sine waves of which a complex tone is composed. A harmonic is any member of the series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The fundamental is also considered a harmonic because it is 1 times itself, a harmonic partial is any real partial component of a complex tone that matches an ideal harmonic. An inharmonic partial is any partial that does not match an ideal harmonic, Inharmonicity is a measure of the deviation of a partial from the closest ideal harmonic, typically measured in cents for each partial. Unpitched, or indefinite-pitched instruments, such as cymbals, gongs, or tam-tams make sounds that are rich in inharmonic partials, an overtone is any partial except the lowest partial. The term overtone does not imply harmonicity or inharmonicity and has no special meaning other than to exclude the fundamental. It is the relative strengths of the different overtones that gives an instrument its particular timbre, some electronic instruments, such as synthesizers, can play a pure frequency with no overtones. Synthesizers can also combine pure frequencies into more complex tones, such as to other instruments. Certain flutes and ocarinas are very nearly without overtones, in most pitched musical instruments, the fundamental is accompanied by other, higher-frequency harmonics. Thus shorter-wavelength, higher-frequency waves occur with varying prominence and give each instrument its characteristic tone quality, the fact that a string is fixed at each end means that the longest allowed wavelength on the string is twice the length of the string. Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, theoretically, these shorter wavelengths correspond to vibrations at frequencies that are 2,3,4,5,6, etc. times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator it vibrates against often alter these frequencies, the harmonic series is an arithmetic series. In terms of frequency, the difference between consecutive harmonics is therefore constant and equal to the fundamental, but because human ears respond to sound nonlinearly, higher harmonics are perceived as closer together than lower ones

2.
Musical scale
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In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is a scale. Some scales contain different pitches when ascending than when descending, for example, the Melodic minor scale. Due to the principle of equivalence, scales are generally considered to span a single octave. A musical scale represents a division of the space into a certain number of scale steps. A measure of the width of each scale step provides a method to classify scales, based on their interval patterns, scales are put into categories including diatonic, chromatic, major, minor, and others. A specific scale is defined by its interval pattern and by a special note. The tonic of a scale is the selected as the beginning of the octave. Typically, the name of the scale specifies both its tonic and its interval pattern, for example, C major indicates a major scale with a C tonic. Scales are typically listed from low to high, most scales are octave-repeating, meaning their pattern of notes is the same in every octave. An octave-repeating scale can be represented as an arrangement of pitch classes. The distance between two notes in a scale is called a scale step. The notes of a scale are numbered by their steps from the root of the scale, for example, in a C major scale the first note is C, the second D, the third E and so on. Two notes can also be numbered in relation to other, C and E create an interval of a third. A single scale can be manifested at many different pitch levels, for example, a C major scale can be started at C4 and ascending an octave to C5, or it could be started at C6, ascending an octave to C7. As long as all the notes can be played, the octave they take on can be altered, the pitch distances or intervals among the notes of a scale tell us more about the sound of the music than does the mere number of tones. The notes of a scale form intervals with each of the notes of the chord in combination. A 5-note scale has 10 of these intervals, a 6-note scale has 15, a 7-note scale has 21

3.
Just intonation
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In music, just intonation or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a pure or just interval, pure intervals are important in music because they correspond to the vibrational patterns found in physical objects which correlate to human perception. The two notes in any just interval are members of the harmonic series. Frequency ratios involving large integers such as 1024,729 are not generally said to be justly tuned, the Indian classical music system uses just intonation tuning as codified in the Natya Shastra. Various societies perceive pure intervals as pleasing or satisfying consonant and, conversely, however, various societies do not have these associations. Just intonation can be contrasted and compared with equal temperament, which dominates Western instruments of fixed pitch, however, except for doubling of frequencies, no other intervals are exact ratios of small integers. Each just interval differs a different amount from its analogous, equally tempered interval, justly tuned intervals can be written as either ratios, with a colon, or as fractions, with a solidus. For example, two tones, one at 300 hertz, and the other at 200 hertz are both multiples of 100 Hz and as members of the harmonic series built on 100 Hz. Thus 3,2, known as a fifth, may be defined as the musical interval between the second and third harmonics of any fundamental pitch. Just intonation An A-major scale, followed by three major triads, and then a progression of fifths in just intonation, equal temperament An A-major scale, followed by three major triads, and then a progression of fifths in equal temperament. By listening to the file, and then listening to this one, one might be able to hear the beating in this file. Equal temperament and just intonation compared A pair of major thirds, the first in each pair is in equal temperament, the second is in just intonation. Equal temperament and just intonation compared with square waveform A pair of major chords, the first is in equal temperament, the second is in just intonation. The pair of chords is repeated with a transition from equal temperament to just temperament between the two chords, in the equal temperament chords a roughness or beating can be heard at about 4 Hz and about 0.8 Hz. In the just intonation triad, this roughness is absent, the square waveform makes the difference between equal and just temperaments more obvious. Harmonic intervals come naturally to horns, vibrating strings, and in human singing voices. Pythagorean tuning, perhaps the first tuning system to be theorized in the West, is a system in all tones can be found using powers of the ratio 3,2. It is easier to think of this system as a cycle of fifths

4.
Limit (music)
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In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony, roughly speaking, the larger the limit number, the more harmonically complex and potentially dissonant will the intervals of the tuning be perceived. A scale belonging to a prime limit has a distinctive hue that makes it aurally distinguishable from scales with other limits. Harry Partch, Ivor Darreg, and Ralph David Hill are among the many microtonalists to suggest that music has been slowly evolving to employ higher and higher harmonics in its constructs, in medieval music, only chords made of octaves and perfect fifths were considered consonant. In the West, triadic harmony arose around the time of the Renaissance, the major and minor thirds of these triads invoke relationships among the first 5 harmonics. Around the turn of the 20th century, tetrads debuted as fundamental building blocks in African-American music, in conventional music theory pedagogy, these seventh chords are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5, for example, the dominant 7th chord in 12-ET approximates 4,5,6,7, while the major 7th chord approximates 8,10,12,15. In just intonation, intervals between pitches are drawn from the rational numbers, since Partch, two distinct formulations of the limit concept have emerged, odd limit and prime limit. Odd limit and prime limit n do not include the same even when n is an odd prime. For a positive odd number n, the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than n. In Genesis of a Music, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partchs theoretical prediction of the dissonance of intervals are very similar to those of theorists including Hermann von Helmholtz, William Sethares. An identity is each of the odd numbers below and including the limit in a tuning, for example, the identities included in 5-limit tuning are 1,3, and 5. The number 9, though not a prime, is nevertheless an identity in music, simply because it is an odd number. Partch defines identity as one of the correlatives, major or minor, in a tonality, one of the odd-number ingredients, odentity and udentity are, short for Over-Identity, and, Under-Identity, respectively. An udentity is an identity of an utonality, for a prime number n, the n-prime-limit contains all rational numbers that can be factored using primes no greater than n. In other words, it is the set of rationals with numerator and denominator both n-smooth, given a prime number p, the subset of Q + consisting of those rational numbers x whose prime factorization has the form x = p 1 α1 p 2 α2. P r ≤ p forms a subgroup of and we say that a scale or system of tuning uses p-limit tuning if all interval ratios between pitches lie in this subgroup

5.
Harmonic
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A harmonic is any member of the harmonic series, the divergent infinite series. Every term of the series after the first is the mean of the neighboring terms. The phrase harmonic mean likewise derives from music, the term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. It is typically applied to repeating signals, such as sinusoidal waves, a harmonic of such a wave is a wave with a frequency that is a positive integer multiple of the frequency of the original wave, known as the fundamental frequency. The original wave is called the 1st harmonic, the following harmonics are known as higher harmonics. As all harmonics are periodic at the frequency, the sum of harmonics is also periodic at that frequency. On strings, harmonics that are bowed have a glassy, pure tone, harmonics may also be called overtones, partials or upper partials. In some music contexts, the harmonic, overtone and partial are used fairly interchangeably. Most acoustic instruments emit complex tones containing many individual partials, rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials. Oscillators that produce harmonic partials behave somewhat like one-dimensional resonators, and are long and thin. Wind instruments whose air column is open at one end, such as trumpets and clarinets. However they only produce partials matching the odd harmonics, at least in theory, the reality of acoustic instruments is such that none of them behaves as perfectly as the somewhat simplified theoretical models would predict. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials, antique singing bowls are known for producing multiple harmonic partials or multiphonics. An overtone is any partial higher than the lowest partial in a compound tone, the relative strengths and frequency relationships of the component partials determine the timbre of an instrument. This chart demonstrates how the three types of names are counted, In many musical instruments, it is possible to play the upper harmonics without the note being present. In a simple case this has the effect of making the note go up in pitch by an octave, in some cases it also changes the timbre of the note. This is part of the method of obtaining higher notes in wind instruments. The extended technique of playing multiphonics also produces harmonics, on string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch

6.
Modulation (music)
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In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature, modulations articulate or create the structure or form of many pieces, as well as add interest. Treatment of a chord as the tonic for less than a phrase is considered tonicization, Modulation is the essential part of the art. Without it there is music, for a piece derives its true beauty not from the large number of fixed modes which it embraces. The modulating dominant is the dominant of the quasi-tonic, the pivot chord is a predominant to the modulating dominant and a chord common to both the keys of the tonic and the quasi-tonic. For example, in a modulation to the dominant, ii/V-V/V-V could be a chord, modulating dominant. For example, G major and D major share 4 chords in common, G, B minor and this can be easily determined by a chart similar to the one below, which compares chord qualities. The I chord in G major—a G major chord—is also the IV chord in D major, so I in G major, any chord with the same root note and chord quality can be used as the pivot chord. However, chords that are not generally found in the style of the piece are not likely to be chosen as the pivot chord. The most common chords are the predominant chords in the new key. In analysis of a piece that uses this style of modulation, an enharmonic modulation takes place when one treats a chord as if it were spelled enharmonically as a functional chord in the destination key, and then proceeds in the destination key. There are two types of enharmonic modulations, dominant seventh/augmented sixth, and diminished seventh. A diminished seventh chord meanwhile, can be respelled in multiple ways to form a diminished seventh chord in a key a minor third. C♯-E-G-B♭, A-C♯-E-G, D-F♯-A takes us to the key of D major - a parallel modulation, enharmonically, C♯-E-G-B♭, A-C♯-E-G = A-C♯-E-G, C♯-E-G♯ modulates to C# minor - a major seventh modulation/half-step descending. C♯-E-G-B♭, C♯-E♭-G-B♭ = E♭-G-B♭-D♭, A♭-C-E♭ leads to A♭ major - a minor third AND relative modulation, in short, lowering any note of a diminished seventh chord a half tone leads to a dominant seventh chord, the lowered note being the root of the new chord. Raising any note of a seventh chord a half tone leads to a half-diminished seventh chord. This means that any diminished chord can be modulated to eight different chords by simply lowering or raising any of its notes and this type of modulation is particularly common in Romantic music, in which chromaticism rose to prominence. Other types of enharmonic modulation include the augmented triad and French sixth, augmented triad modulation occurs in the same fashion as the diminished seventh, that is, to modulate to another augmented triad in a key, a major third or minor sixth away

7.
Synthesizer
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A synthesizer is an electronic musical instrument that generates electric signals that are converted to sound through instrument amplifiers and loudspeakers or headphones. Synthesizers may either imitate instruments like piano, Hammond organ, flute, vocals, natural sounds like ocean waves, etc. or generate new electronic timbres. Synthesizers without built-in controllers are called sound modules, and are controlled via USB, MIDI or CV/gate using a controller device. Synthesizers use various methods to generate electronic signals, synthesizers were first used in pop music in the 1960s. In the 1970s, synths were used in disco, especially in the late 1970s, in the 1980s, the invention of the relatively inexpensive, mass market Yamaha DX7 synth made synthesizers widely available. 1980s pop and dance music often made use of synthesizers. In the 2010s, synthesizers are used in genres of pop, rock. Contemporary classical music composers from the 20th and 21st century write compositions for synthesizer, the beginnings of the synthesizer are difficult to trace, as it is difficult to draw a distinction between synthesizers and some early electric or electronic musical instruments. One of the earliest electric musical instruments, the telegraph, was invented in 1876 by American electrical engineer Elisha Gray. He accidentally discovered the sound generation from a self-vibrating electromechanical circuit and this musical telegraph used steel reeds with oscillations created by electromagnets transmitted over a telegraph line. Gray also built a simple loudspeaker device into later models, consisting of a diaphragm in a magnetic field. This instrument was a remote electromechanical musical instrument that used telegraphy, though it lacked an arbitrary sound-synthesis function, some have erroneously called it the first synthesizer. In 1897, Thaddeus Cahill invented the Teleharmonium, which used dynamos, and was capable of additive synthesis like the Hammond organ, however, Cahills business was unsuccessful for various reasons, and similar but more compact instruments were subsequently developed, such as electronic and tonewheel organs. In 1906, American engineer, Lee De Forest ushered in the electronics age and he invented the first amplifying vacuum tube, called the Audion tube. This led to new entertainment technologies, including radio and sound films, ondes Martenot and Trautonium were continuously developed for several decades, finally developing qualities similar to later synthesizers. In the 1920s, Arseny Avraamov developed various systems of graphic sonic art, in 1938, USSR engineer Yevgeny Murzin designed a compositional tool called ANS, one of the earliest real-time additive synthesizers using optoelectronics. The earliest polyphonic synthesizers were developed in Germany and the United States, during the three years that Hammond manufactured this model,1,069 units were shipped, but production was discontinued at the start of World War II. Both instruments were the forerunners of the electronic organs and polyphonic synthesizers

8.
Transposition (music)
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In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval. For example, one might transpose a piece of music into another key. Similarly, one might transpose a tone row or a collection of pitches such as a chord so that it begins on another pitch. The transposition of a set A by n semitones is designated by Tn, thus the set consisting of 0–1–2 transposed by 5 semitones is 5–6–7 since 0 +5 =5,1 +5 =6, and 2 +5 =7. There are two different kinds of transposition, depending on one is measuring intervals according to the chromatic scale or some other scale. In chromatic transposition one shifts every pitch in a collection of notes by a number of semitones. For instance, if one transposes the pitches C4–E4–G4 upwards by four semitones, in scalar transposition one shifts every pitch in a collection by a fixed number of scale steps relative to some scale. For example, if one transposes the pitches C4–E4–G4 up by two steps relative to the familiar C major scale, one obtains the pitches E4–G4–B4, if one transposes the same pitches up by two steps relative to the F major scale, one obtains instead E4–G4–B♭4. Scalar transposition is sometimes called diatonic transposition, but this term can be misleading, however, scalar transposition can occur with respect to any type of scale, not just the diatonic. There are two kinds of transposition, by pitch interval or by pitch interval class, applied to pitches or pitch classes. Transposition may be applied to pitches or to pitch classes, although transpositions are usually written out, musicians are occasionally asked to transpose music at sight, that is, to read the music in one key while playing in another. There are three basic techniques for teaching sight transposition, interval, clef, and numbers, first one determines the interval between the written key and the target key. Then one imagines the notes up by the corresponding interval, a performer using this method may calculate each note individually, or group notes together. Clef transposition is routinely taught in Belgium and France, one imagines a different clef and a different key signature than the ones printed. The change of clef is used so that the lines and spaces correspond to different notes than the lines and spaces of the original score. Seven clefs are used for this, treble, bass, baritone, and C-clefs on the four lowest lines, the signature is then adjusted for the actual accidental one wants on that note. The octave may also have to be adjusted, but this is a matter for most musicians. Transposing by numbers means, one determines the degree of the written note in the given key

9.
Fundamental frequency
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The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the frequency is the lowest frequency sinusoidal in the sum. In some contexts, the fundamental is usually abbreviated as f0, in other contexts, it is more common to abbreviate it as f1, the first harmonic. All sinusoidal and many non-sinusoidal waveforms are periodic, which is to say they repeat exactly over time. The period of a waveform is the T for which the equation is true. This means that this equation and a definition of the values over any interval of length T is all that is required to describe the waveform completely. Every waveform may be described using any multiple of this period, there exists a smallest period over which the function may be described completely and this period is the fundamental period. The fundamental frequency is defined as its reciprocal, f 0 =1 T Since the period is measured in units of time, when the time units are seconds, the frequency is in s −1, also known as Hertz. For a tube of length L with one end closed and the end open the wavelength of the fundamental harmonic is 4 L. If the ends of the tube are now both closed or both opened as in the last two animations, the wavelength of the fundamental harmonic becomes 2 L. By the same method as above, the frequency is found to be f 0 = v 2 L. At 20 °C the speed of sound in air is 343 m/s and this speed is temperature dependent and does increase at a rate of 0.6 m/s for each degree Celsius increase in temperature. The fundamental may be created by vibration over the length of a string or air column. The fundamental is one of the harmonics, a harmonic is any member of the harmonic series, an ideal set of frequencies that are positive integer multiples of a common fundamental frequency. The reason a fundamental is also considered a harmonic is because it is 1 times itself, the fundamental is the frequency at which the entire wave vibrates. Overtones are other components present at frequencies above the fundamental. All of the components that make up the total waveform, including the fundamental

10.
Beauty in the Beast
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The album uses alternate musical tunings and scales, influenced by jazz and world music. On the back she includes a quote by Van Gogh, I am always doing what I cannot do yet, as the liner notes state, the entire album is synthesized, meaning that All the music and sounds heard on this recording were directly digitally generated. This eliminates all the limitations of microphones, the weak link necessary in all other digital recordings. A notable exception is Wendy Carlos, who has composed a deal of music for synthesizers using many different scales. Particularly recommend Beauty in the Beast, the music on this album cuts through a lot of the conventions and restraints that were used as frameworks for previous releases, instrumentation, tonality, and scaling, to name just a few

11.
Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, another way of saying that a integer is a square number, is that its square root is again an integer. For example, √9 =3, so 9 is a square number, a positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, the concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two integers, and, conversely, the ratio of two square integers is a square, e. g.49 =2. Starting with 1, there are ⌊√m⌋ square numbers up to and including m, the squares smaller than 602 =3600 are, The difference between any perfect square and its predecessor is given by the identity n2 −2 = 2n −1. Equivalently, it is possible to count up square numbers by adding together the last square, the last squares root, and the current root, that is, n2 =2 + + n. The number m is a number if and only if one can compose a square of m equal squares. Hence, a square with side length n has area n2, the expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, the formula follows, n 2 = ∑ k =1 n. So for example,52 =25 =1 +3 +5 +7 +9, there are several recursive methods for computing square numbers. For example, the nth square number can be computed from the square by n2 =2 + + n =2 +. Alternatively, the nth square number can be calculated from the two by doubling the th square, subtracting the th square number, and adding 2. For example, 2 × 52 −42 +2 = 2 × 25 −16 +2 =50 −16 +2 =36 =62, a square number is also the sum of two consecutive triangular numbers. The sum of two square numbers is a centered square number. Every odd square is also an octagonal number

12.
Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose two. Carl Friedrich Gauss is said to have found this relationship in his early youth, however, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus, the triangular number Tn solves the handshake problem of counting the number of handshakes if each person in a room with n +1 people shakes hands once with each person. In other words, the solution to the problem of n people is Tn−1. The function T is the analog of the factorial function. In the limit, the ratio between the two numbers, dots and line segments is lim n → ∞ T n L n =13, Triangular numbers have a wide variety of relations to other figurate numbers. Most simply, the sum of two triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically, T n + T n −1 = + = + = n 2 =2, alternatively, the same fact can be demonstrated graphically, There are infinitely many triangular numbers that are also square numbers, e. g.1,36,1225. Some of them can be generated by a recursive formula. All square triangular numbers are found from the recursion S n =34 S n −1 − S n −2 +2 with S0 =0 and S1 =1. Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n and this can also be expressed as ∑ k =1 n k 3 =2. The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number, more generally, the difference between the nth m-gonal number and the nth -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the triangular number,15. Every other triangular number is a hexagonal number, knowing the triangular numbers, one can reckon any centered polygonal number, the nth centered k-gonal number is obtained by the formula C k n = k T n −1 +1 where T is a triangular number. The positive difference of two numbers is a trapezoidal number. Triangular numbers correspond to the case of Faulhabers formula. Alternating triangular numbers are also hexagonal numbers, every even perfect number is triangular, given by the formula M p 2 p −1 = M p 2 = T M p where Mp is a Mersenne prime

13.
Note (music)
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In music, the term note has three primary meanings, A sign used in musical notation to represent the relative duration and pitch of a sound, A pitched sound itself. Notes are the blocks of much written music, discretizations of musical phenomena that facilitate performance, comprehension. In the former case, one note to refer to a specific musical event, in the latter. Two notes with fundamental frequencies in an equal to any integer power of two are perceived as very similar. Because of that, all notes with these kinds of relations can be grouped under the pitch class. However, within the English-speaking and Dutch-speaking world, pitch classes are represented by the first seven letters of the Latin alphabet. A few European countries, including Germany, adopt an almost identical notation, the eighth note, or octave, is given the same name as the first, but has double its frequency. The name octave is also used to indicate the span between a note and another with double frequency, for example, the now-standard tuning pitch for most Western music,440 Hz, is named a′ or A4. There are two systems to define each note and octave, the Helmholtz pitch notation and the scientific pitch notation. Letter names are modified by the accidentals, a sharp ♯ raises a note by a semitone or half-step, and a flat ♭ lowers it by the same amount. In modern tuning a half step has a ratio of 12√2. The accidentals are written after the name, so, for example, F♯ represents F-sharp, B♭ is B-flat. Additional accidentals are the double-sharp, raising the frequency by two semitones, and double-flat, lowering it by that amount, in musical notation, accidentals are placed before the note symbols. Systematic alterations to the seven lettered pitches in the scale can be indicated by placing the symbols in the key signature, explicitly noted accidentals can be used to override this effect for the remainder of a bar. A special accidental, the natural symbol ♮, is used to indicate an unmodified pitch, effects of key signature and local accidentals do not accumulate. If the key signature indicates G♯, a flat before a G makes it G♭, though often this type of rare accidental is expressed as a natural. Likewise, a sharp sign on a key signature with a single sharp ♯ indicates only a double sharp. Assuming enharmonicity, many accidentals will create equivalences between pitches that are written differently, for instance, raising the note B to B♯ is equal to the note C

14.
Octave
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In music, an octave or perfect octave is the interval between one musical pitch and another with half or double its frequency. It is defined by ANSI as the unit of level when the base of the logarithm is two. The octave relationship is a phenomenon that has been referred to as the basic miracle of music. The most important musical scales are written using eight notes. For example, the C major scale is typically written C D E F G A B C, two notes separated by an octave have the same letter name and are of the same pitch class. Three commonly cited examples of melodies featuring the perfect octave as their opening interval are Singin in the Rain, Somewhere Over the Rainbow, the interval between the first and second harmonics of the harmonic series is an octave. The octave has occasionally referred to as a diapason. To emphasize that it is one of the intervals, the octave is designated P8. The octave above or below a note is sometimes abbreviated 8a or 8va, 8va bassa. For example, if one note has a frequency of 440 Hz, the note one octave above is at 880 Hz, the ratio of frequencies of two notes an octave apart is therefore 2,1. Further octaves of a note occur at 2n times the frequency of that note, such as 2,4,8,16, etc. and the reciprocal of that series. For example,55 Hz and 440 Hz are one and two away from 110 Hz because they are 1⁄2 and 4 times the frequency, respectively. After the unison, the octave is the simplest interval in music, the human ear tends to hear both notes as being essentially the same, due to closely related harmonics. Notes separated by a ring together, adding a pleasing sound to music. For this reason, notes an octave apart are given the note name in the Western system of music notation—the name of a note an octave above A is also A. The conceptualization of pitch as having two dimensions, pitch height and pitch class, inherently include octave circularity, thus all C♯s, or all 1s, in any octave are part of the same pitch class. Octave equivalency is a part of most advanced cultures, but is far from universal in primitive. The languages in which the oldest extant written documents on tuning are written, leon Crickmore recently proposed that The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week

15.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

16.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain content is freely available to anyone. William G. Bowen, president of Princeton University from 1972 to 1988, JSTOR originally was conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term, online access and full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution, JSTOR was initiated in 1995 at seven different library sites, and originally encompassed ten economics and history journals. JSTOR access improved based on feedback from its sites. Special software was put in place to make pictures and graphs clear, with the success of this limited project, Bowen and Kevin Guthrie, then-president of JSTOR, wanted to expand the number of participating journals. They met with representatives of the Royal Society of London and an agreement was made to digitize the Philosophical Transactions of the Royal Society dating from its beginning in 1665, the work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially, until January 2009 JSTOR operated as an independent, self-sustaining nonprofit organization with offices in New York City and in Ann Arbor, Michigan. JSTOR content is provided by more than 900 publishers, the database contains more than 1,900 journal titles, in more than 50 disciplines. Each object is identified by an integer value, starting at 1. In addition to the site, the JSTOR labs group operates an open service that allows access to the contents of the archives for the purposes of corpus analysis at its Data for Research service. This site offers a facility with graphical indication of the article coverage. Users may create focused sets of articles and then request a dataset containing word and n-gram frequencies and they are notified when the dataset is ready and may download it in either XML or CSV formats. The service does not offer full-text, although academics may request that from JSTOR, JSTOR Plant Science is available in addition to the main site. The materials on JSTOR Plant Science are contributed through the Global Plants Initiative and are only to JSTOR

17.
Musical tuning
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In music, there are two common meanings for tuning, Tuning practice, the act of tuning an instrument or voice. Tuning systems, the systems of pitches used to tune an instrument. Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones, Tuning is usually based on a fixed reference, such as A =440 Hz. Out of tune refers to a pitch/tone that is too high or too low in relation to a given reference pitch. While an instrument might be in relative to its own range of notes. Some instruments become out of tune with damage or time and must be readjusted or repaired, different methods of sound production require different methods of adjustment, Tuning to a pitch with ones voice is called matching pitch and is the most basic skill learned in ear training. Turning pegs to increase or decrease the tension on strings so as to control the pitch, instruments such as the harp, piano, and harpsichord require a wrench to turn the tuning pegs, while others such as the violin can be tuned manually. Modifying the length or width of the tube of an instrument, brass instrument, pipe, bell. The sounds of instruments such as cymbals are inharmonic—they have irregular overtones not conforming to the harmonic series. Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other, a tuning fork or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals often a piano is used. Symphony orchestras and concert bands tend to tune to an A or a B♭, respectively, interference beats are used to objectively measure the accuracy of tuning. As the two approach a harmonic relationship, the frequency of beating decreases. When tuning a unison or octave it is desired to reduce the beating frequency until it cannot be detected, for other intervals, this is dependent on the tuning system being used. Harmonics may be used to facilitate tuning of strings that are not themselves tuned to the unison, for example, lightly touching the highest string of a cello at the middle while bowing produces the same pitch as doing the same a third of the way down its second-highest string. The resulting unison is more easily and quickly judged than the quality of the fifth between the fundamentals of the two strings. In music, the open string refers to the fundamental note of the unstopped. The strings of a guitar are tuned to fourths, as are the strings of the bass guitar. Violin, viola, and cello strings are tuned to fifths, however, non-standard tunings exist to change the sound of the instrument or create other playing options

18.
Pitch (music)
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Pitch can be determined only in sounds that have a frequency that is clear and stable enough to distinguish from noise. Pitch is a major attribute of musical tones, along with duration, loudness. Pitch may be quantified as a frequency, but pitch is not a purely objective physical property, Pitch is an auditory sensation in which a listener assigns musical tones to relative positions on a musical scale based primarily on their perception of the frequency of vibration. Pitch is closely related to frequency, but the two are not equivalent, frequency is an objective, scientific attribute that can be measured. Pitch is each persons subjective perception of a wave, which cannot be directly measured. However, this not necessarily mean that most people wont agree on which notes are higher and lower. Sound waves themselves do not have pitch, but their oscillations can be measured to obtain a frequency and it takes a sentient mind to map the internal quality of pitch. However, pitches are usually associated with, and thus quantified as frequencies in cycles per second, or hertz, by comparing sounds with pure tones, Complex and aperiodic sound waves can often be assigned a pitch by this method. According to the American National Standards Institute, pitch is the attribute of sound according to which sounds can be ordered on a scale from low to high. That is, high pitch means very rapid oscillation, and low pitch corresponds to slower oscillation, despite that, the idiom relating vertical height to sound pitch is shared by most languages. At least in English, it is just one of many deep conceptual metaphors that involve up/down, the exact etymological history of the musical sense of high and low pitch is still unclear. There is evidence that humans do actually perceive that the source of a sound is slightly higher or lower in vertical space when the frequency is increased or reduced. The pitch of tones can be ambiguous, meaning that two or more different pitches can be perceived, depending upon the observer. In a situation like this, the percept at 200 Hz is commonly referred to as the missing fundamental, Pitch depends to a lesser degree on the sound pressure level of the tone, especially at frequencies below 1,000 Hz and above 2,000 Hz. The pitch of lower tones gets lower as sound pressure increases, for instance, a tone of 200 Hz that is very loud seems one semitone lower in pitch than if it is just barely audible. Above 2,000 Hz, the pitch gets higher as the sound gets louder, theories of pitch perception try to explain how the physical sound and specific physiology of the auditory system work together to yield the experience of pitch. In general, pitch perception theories can be divided into place coding, place theory holds that the perception of pitch is determined by the place of maximum excitation on the basilar membrane. However, a purely place-based theory cannot account for the accuracy of pitch perception in the low, temporal theories offer an alternative that appeals to the temporal structure of action potentials, mostly the phase-locking and mode-locking of action potentials to frequencies in a stimulus

19.
Cent (music)
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The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each, alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquets suggestion. It has become the method of representing and comparing musical pitches. Like a decibels relation to intensity, a cent is a ratio between two close frequencies, for the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies. An equally tempered semitone spans 100 cents by definition, an octave—two notes that have a frequency ratio of 2, 1—spans twelve semitones and therefore 1200 cents.0005777895. For example, in just intonation the major third is represented by the frequency ratio 5,4, applying the formula at the top shows that this is about 386 cents. The equivalent interval on the piano would be 400 cents. The difference,14 cents, is about a seventh of a half step, as x increases from 0 to 1⁄12, the function 2x increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a linear function that is numerically correct at semitones. That is, n cents for n from 0 to 100 may be approximated as 1 +0. 0005946n instead of 2 n⁄1200. The rounded error is zero when n is 0 or 100, and is about 0.72 cents high when n is 50 and this error is well below anything humanly audible, making this piecewise linear approximation adequate for most practical purposes. It is difficult to establish how many cents are perceptible to humans, one author stated that humans can distinguish a difference in pitch of about 5–6 cents. The threshold of what is perceptible, technically known as the just noticeable difference, also varies as a function of the frequency, the amplitude and the timbre. In one study, changes in tone quality reduced student musicians ability to recognize, as out-of-tune and it has also been established that increased tonal context enables listeners to judge pitch more accurately. Free, online web sites for self-testing are available, while intervals of less than a few cents are imperceptible to the human ear in a melodic context, in harmony very small changes can cause large changes in beats and roughness of chords. When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch, normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents, iring noticed that the Grad/Werckmeister and the schisma are nearly the same and both may be approximated by 600 steps per octave. Yasser promoted the decitone, centitone, and millitone, for example, Equal tempered perfect fifth =700 cents =175.6 savarts =583.3 millioctaves =350 centitones

20.
Millioctave
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The millioctave is a unit of measurement for musical intervals. As is expected from the prefix milli-, a millioctave is defined as 1/1000 of an octave, from this it follows that one millioctave is equal to the ratio 21/1000, the 1000th root of 2, or approximately 1.0006934. A millioctave is exactly 1.2 cents, the millioctave was introduced by the German physicist Arthur von Oettingen in his book Das duale Harmoniesystem. The invention goes back to John Herschel, who proposed a division of the octave into 1000 parts, compared to the cent, the millioctave has not been as popular. It is, however, occasionally used by authors who wish to avoid the association between the cent and equal temperament. However, it has criticized that it introduces a bias for the less familiar 10-tone equal temperament. Cent Savart Musical tuning Logarithm Degree Chiliagon Logarithmic Interval Measures

21.
Savart
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The savart /səˈvɑːr/ is a unit of measurement for musical pitch intervals. One savart is equal to one thousandth of a decade,3.9863 cents, musically, in just intonation, the interval of a decade is precisely a just major twenty-fourth, or, in other words, three octaves and a just major third. Today the savart has largely replaced by the cent and the millioctave. The savart is practically the same as the earlier heptameride, one seventh of a meride, one tenth of an heptameride is a decameride and a hundredth of an heptameride is a jot. The number of savarts in an octave is 1000 times the logarithm of 2. Sometimes this is rounded to 300, which makes the unit more useful for equal temperament, Sauveur proposed the méride, eptaméride, and decaméride. In English these are meride, heptameride, and decameride respectively, the octave is divided into 43 merides, the meride is divided into seven heptamerides, and the heptameride is divided into ten decamerides. There are thus 43 ×7 =301 heptamerides in an octave. The attraction of this scheme to Sauveur was that log10 is very close to.301 and this is equivalent to assuming 1000 heptamerides in a decade rather than 301 in an octave, the same as Savarts definition. The unit was given the name savart sometime in the 20th century, a disadvantage of this scheme is that there are not an exact number of heptamerides/savarts in an equal tempered semitone. For this reason Alexander Wood used a definition of the savart, with 300 savarts in an octave. A related unit is the jot, of which there are 30103 in an octave, the jot is defined in a similar way to the savart, but has a more accurate rounding of log10 because more digits are used. There are approximately 100 jots in a savart, the unit was first described by Augustus de Morgan which he called an atom. The name jot was coined by John Curwen at the suggestion of Hermann von Helmholtz

22.
Interval (music)
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In music theory, an interval is the difference between two pitches. In Western music, intervals are most commonly differences between notes of a diatonic scale, the smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones and they can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, Intervals can be arbitrarily small, and even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic frequencies, for example, any two notes an octave apart have a frequency ratio of 2,1. This means that successive increments of pitch by the same result in an exponential increase of frequency. For this reason, intervals are often measured in cents, a derived from the logarithm of the frequency ratio. In Western music theory, the most common naming scheme for intervals describes two properties of the interval, the quality and number, examples include the minor third or perfect fifth. These names describe not only the difference in semitones between the upper and lower notes, but also how the interval is spelled, the importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭. The size of an interval can be represented using two alternative and equivalently valid methods, each appropriate to a different context, frequency ratios or cents, the size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, Intervals with small-integer ratios are often called just intervals, or pure intervals. Most commonly, however, musical instruments are tuned using a different tuning system. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, for instance, an equal-tempered fifth has a frequency ratio of 2 7⁄12,1, approximately equal to 1.498,1, or 2.997,2. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems, the standard system for comparing interval sizes is with cents. The cent is a unit of measurement. If frequency is expressed in a scale, and along that scale the distance between a given frequency and its double is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament, a system in which all semitones have the same size. Hence, in 12-TET the cent can be defined as one hundredth of a semitone

23.
Interval ratio
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In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3,2,1.5, if the A above middle C is 440 Hz, the perfect fifth above it would be E, at 660 Hz, while the equal tempered E5 is 659.255 Hz. Ratios have a relationship to string length, for example stopping a string at two-thirds its length produces a pitch one. Intervals may be ranked by relative consonance and dissonance, as such ratios with lower integers are generally more consonant than intervals with higher integers. For example,2,1,4,3,9,8,65536,59049, consonance and dissonance may more subtly be defined by limit, wherein the ratios whose limit, which includes its integer multiples, is lower are generally more consonant. For example, the 3-limit 128,81 and the 7-limit 14,9, despite having larger integers 128,81 is less dissonant than 14,9, as according to limit theory. For ease of comparison intervals may also be measured in cents, for example, the just perfect fifth is 701.955 cents while the equal tempered perfect fifth is 700 cents. Frequency ratios are used to describe intervals in both Western and non-Western music. When a musical instrument is tuned using a just intonation tuning system, intervals with small-integer ratios are often called just intervals, or pure intervals. To most people, just intervals sound consonant, i. e. pleasant, although the size of equally tuned intervals is typically similar to that of just intervals, in most cases it cannot be expressed by small-integer ratios. For instance, a tempered perfect fifth has a frequency ratio of about 1.4983,1. For a comparison between the size of intervals in different tuning systems, see section Size in different tuning systems

24.
Pitch class
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In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e. g. the pitch class C consists of the Cs in all octaves. The pitch class C stands for all possible Cs, in whatever octave position, important to musical set theory, a pitch class is, all pitches related to each other by octave, enharmonic equivalence, or both. Psychologists refer to the quality of a pitch as its chroma, a chroma is an attribute of pitches, just like hue is an attribute of color. A pitch class is a set of all pitches that share the same chroma, because octave-related pitches belong to the same class, when an octave is reached, the numbers begin again at zero. Indeed, the mapping from pitch to real numbers defined in this forms the basis of the MIDI Tuning Standard. To represent pitch classes, we need to identify or glue together all pitches belonging to the same pitch class—i. e, all numbers p and p +12. The result is a quotient group that musicians call pitch class space. Points in this space can be labelled using real numbers in the range 0 ≤ x <12. These numbers provide numerical alternatives to the names of elementary music theory,0 = C,1 = C♯/D♭,2 = D,2.5 = D,3 = D♯/E♭. In this system, pitch classes represented by integers are classes of equal temperament. In music, integer notation is the translation of pitch classes and/or interval classes into whole numbers, thus if C =0, then C♯ =1. A♯ =10, B =11, with 10 and 11 substituted by t and e in some sources and this allows the most economical presentation of information regarding post-tonal materials. In the integer model of pitch, all classes and intervals between pitch classes are designated using the numbers 0 through 11. It is not used to music for performance, but is a common analytical and compositional tool when working with chromatic music, including twelve tone, serial. The C above this is not 12, but 0 again, thus arithmetic modulo 12 is used to represent octave equivalence. One advantage of this system is that it ignores the spelling of notes according to their diatonic functionality, there are a few disadvantages with integer notation. First, theorists have used the same integers to indicate elements of different tuning systems. Thus, the numbers 0,1,2,5, are used to notate pitch classes in 6-tone equal temperament

25.
Consonance and dissonance
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In music, consonance and dissonance form a structural dichotomy in which the terms define each other by mutual exclusion, a consonance is what is not dissonant, and reciprocally. However, a finer consideration shows that the forms a gradation. Consonance is associated with sweetness, pleasantness and acceptability and dissonance with harshness, unpleasantness, as Hindemith stressed, The two concepts have never been completely explained, and for a thousand years the definitions have varied. The opposition can be made in different contexts, In acoustics or psychophysiology, in modern times, it usually is based on the perception of harmonic partials of the sounds considered, to such an extent that the distinction really holds only in the case of harmonic sounds. In music, even if the opposition often is founded on the preceding, objective distinction, it often is subjective, conventional, cultural. A major second would be considered dissonant if it occurred in a J. S, Bach prelude from the 1700s, however, the same interval may sound consonant in the context of a Claude Debussy piece from the early 1900s or an atonal contemporary piece. For this reason, consonance and dissonance have been considered particularly in the case of Western polyphonic music, most historical definitions of consonance and dissonance since about the 16th century have stressed their pleasant/unpleasant, or agreeable/disagreeable character. In addition, the oppositions pleasant/unpleasant or agreeable/disagreeable evidence a confusion between the concepts of dissonance and of noise and these include, Frequency ratios, with ratios of lower simple numbers being more consonant than those that are higher. Many of these definitions do not require exact integer tunings, only approximation, coincidence of partials, with consonance being a greater coincidence of partials. By this definition, consonance is dependent not only on the width of the interval between two notes, but also on the spectral distribution and thus sound quality of the notes. Thus, a note and the note one octave higher are highly consonant because the partials of the note are also partials of the lower note. Although Helmholtzs work focused almost exclusively on harmonic timbres and also the tunings, subsequent work has generalized his findings to embrace non-harmonic tunings, fusion, perception of unity or tonal fusion between two notes. A stable tone combination is a consonance, consonances are points of arrival, rest, an unstable tone combination is a dissonance, its tension demands an onward motion to a stable chord. Thus dissonant chords are active, traditionally they have been considered harsh and have expressed pain, grief, in Western music, dissonance is the quality of sounds that seems unstable and has an aural need to resolve to a stable consonance. Both consonance and dissonance are words applied to harmony, chords, and intervals and, by extension, to melody, tonality, and even rhythm and metre. Nevertheless, the ideas of dissonance, consonance, and resolution exist in some form in all musical traditions that have a concept of melody, harmony. Dissonance being the complement of consonance it may be defined, as above, as non-coincidence of partials, lack of fusion or pattern matching, or as complexity. For many musicians and composers, the ideas of dissonance

26.
List of pitch intervals
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Below is a list of intervals exprimable in terms of a prime limit, completed by a choice of intervals in various equal subdivisions of the octave or of other intervals. For instance, the limit of the just perfect fourth is 3, there exists another type of limit, the odd limit, a concept used by Harry Partch, but it is not used here. The term limit was devised by Partch, by definition, every interval in a given limit can also be part of a limit of higher order. For instance, a 3-limit unit can also be part of a 5-limit tuning, by sorting the limit columns in the table below, all intervals of a given limit can be brought together. Pythagorean tuning means 3-limit intonation—a ratio of numbers with prime factors no higher than three, just intonation means 5-limit intonation—a ratio of numbers with prime factors no higher than five. Septimal, undecimal, tridecimal, and septendecimal mean, respectively,7,11,13, meantone refers to meantone temperament, where the whole tone is the mean of the major third. In a meantone temperament, each fifth is narrowed by the small amount. The music program Logic Pro uses also 1/2-comma meantone temperament, equal-tempered refers to X-tone equal temperament with intervals corresponding to X divisions per octave. Tempered intervals however cannot be expressed in terms of prime limits and, the table can also be sorted by frequency ratio, by cents, or alphabetically. List of chord progressions List of meantone intervals List of musical scales and modes Names of seven-limit commas, XenHarmony. org

27.
List of intervals in 5-limit just intonation
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The intervals of 5-limit just intonation are ratios involving only the powers of 2,3, and 5. The fundamental intervals are the superparticular ratios 2/1, 3/2 and 5/4 and that is, the notes of the major triad are in the ratio 1, 5/4, 3/2 or 4,5,6. In all tunings, the third is equivalent to two major seconds. However, because just intonation does not allow the irrational ratio of √5/2, the intervals within the diatonic scale are shown in the table below. The table below shows how these steps map to the first 31 scientific harmonics, § These intervals also appear in the upper table, although with different ratios. List of musical intervals List of pitch intervals Pythagorean interval

28.
Microtonal music
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Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, which are also called microintervals. It may also be extended to any music using intervals not found in the customary Western tuning of twelve equal intervals per octave. Microtonal music can refer to any music containing microtones, therefore it is important to comprehend what a microtone is, the words microtone and microtonal were coined before 1912 by Maud MacCarthy Mann in order to avoid the misnomer quarter tone when speaking of the srutis of Indian music. Prior to this time the quarter tone was used, confusingly, not only for an interval actually half the size of a semitone. 1998, Wallon 1980,13, Whitfield 1989,13, microinterval is a frequent alternative in English, especially in translations of writings by French authors and in discussion of music by French composers. In English, the two terms microtone and microinterval are synonymous, microtone is also sometimes used to refer to individual notes, microtonal pitches added to and distinct from the familiar twelve notes of the chromatic scale, as enharmonic microtones, for example. In English the word microtonality is mentioned in 1946 by Rudi Blesh who related it to microtonal inflexions of the blues scales. It was used earlier by W. McNaught with reference to developments in modernism in a 1939 record review of the Columbia History of Music. The term microinterval is used alongside microtone by American musicologist Margo Schulter in her articles on medieval music, the term microtonal music usually refers to music containing very small intervals but can include any tuning that differs from Western twelve-tone equal temperament. Microtonal variation of intervals is standard practice in the African-American musical forms of spirituals, blues, many microtonal equal divisions of the octave have been proposed, usually in order to achieve approximation to the intervals of just intonation. Terminology other than microtonal has been used or proposed by some theorists, in 1914, A. H. Fox Strangways objected that heterotone would be a better name for śruti than the usual translation microtone. Modern Indian researchers yet write, microtonal intervals called shrutis, a similar term, subchromatic, has been used recently by theorist Marek Žabka. Ivor Darreg proposed the term xenharmonic, another Russian authors use more international adjective microtonal and rendered it in Russian as микротоновый, but not microtonality. However are used микротональность and микротоника also, some authors writing in French have adopted the term micro-intervallique to describe such music. Italian musicologist Luca Conti dedicated two his monographs to microtonalismo, which is the term in Italian, and also in Spanish. The analogous English form, microtonalism, is found occasionally instead of microtonality. At the time when serialism and neoclassicism were still incipient a third movement emerged, the term macrotonal has also been used for musical form. The Hellenic civilizations of ancient Greece left fragmentary records of their music—e. g, ancient Greek intervals were of many different sizes, including microtones

29.
Harry Partch's 43-tone scale
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The 43-tone scale is a just intonation scale with 43 pitches in each octave, invented and used by Harry Partch. Almost all of Partchs music is written in the 43-tone scale, Partch chose the 11 limit as the basis of his music, because the 11th harmonic is the first that is utterly foreign to Western ears. Here are all the ratios within the octave with odd factors up to and including 11, note that the inversion of every interval is also present, so the set is symmetric about the octave. There are two reasons why the 11-limit ratios by themselves would not make a good scale, first, the scale only contains a complete set of chords based on one tonic pitch. Second, it contains large gaps, between the tonic and the two pitches to either side, as well as other places. Both problems can be solved by filling in the gaps with multiple-number ratios, together with the 29 ratios of the 11 limit, these 14 multiple-number ratios make up the full 43-tone scale. Erv Wilson who worked with Partch has pointed out that these added tones form a constant structure of 41 tones with two variables, a constant structure giving one the property of anytime a ratio appears it will be subtended by the same amount of steps. In this way Partch resolved his harmonic and melodic symmetry in one of the best ways possible, the 43-tone scale was published in Genesis of a Music, and is sometimes known as the Genesis scale, or Partchs pure scale. Besides the 11-limit diamond, he also published 5- and 13-limit diamonds, erv Wilson who did the original drawings in Partchs Genesis of a Music has made a series of diagrams of Partchs diamond as well as others like Diamonds

30.
Hexany
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In music theory, the hexany is a six-note just intonation structure, with the notes placed on the vertices of an octahedron, equivalently the faces of a cube. The notes are arranged so that edge of the octahedron joins together notes that make a consonant dyad. This makes a musical geometry with the form of the octahedron. The edges of the octahedron show musical intervals between the vertices, usually chosen to be consonant intervals from the harmonic series, the points represent musical notes and the three notes that make each of the triangular faces represent musical triads. Its constructed by taking four musical intervals, one of which can optionally be the unison, and then combining them in pairs, in all possible ways. So for instance if you start with 1/1, 3/1, 5/1 and 7/1 then combine them in pairs you get 1*3, 1*5, 1*7, 3*5, 3*7, 5*7 and those are the notes of the 1,3,5,7 hexany. The notes are often octave shifted to them all within the same octave, which has no effect on interval relations. The 1,3,5,7 hexany is found within any 3D cubic lattice of musical pitches, and so within the three factor Euler–Fokker genus based on a cube. If none of the used to construct it is the unity then you need to go into four dimensions. An example of this is the 3,5,7,11 hexany, the result is still a three dimensional figure, the octahedron, with vertices 3*5, 3*7, 3*11, 5*7, 5*11, 7*11. However when you embed it in the four factor Euler Fokker genus and then represent this in 4D and you can have 3D cross sections of a 4D shape much as you can obtain a triangle as a 2D cross section of a normal 3D cube. The Hexany is the invention of Erv Wilson and represents one of the simplest structures found in his Combination Product Sets, the numbers of vertices of his combination sets when set out as subdivisions of a Euler-Fokker genus follow the numbers in Pascals triangle. IN this construction, the hexany is the cross section of the four factor genus. Hexany is the name that Erv Wilson gave to the six notes in the 2-out-of-4 combination product set 4 CPS) and this shows the three dimensional version of the hexany. Here is another diagram showing how the hexany can be found in the three factor Euler Fokker genus, the hexany is the figure containing both the triangles shown as well as the connecting lines between them. Note - in this 3D construction it is not visually a perfect octahedron - it is somewhat squashed, but that is an inessential difference as the interval relationships are the same. See also figure 2 of Kraig Gradys paper, in its most general form the hexany is embedded in a four factor Euler–Fokker genus, or geometrically, a hypercube, also called a tesseract. The four dimensions of the hypercube are often tuned to distinct primes to achieve a hexany with maximally consonant triads, a single step in each dimension corresponds to multiplying the frequency by that prime

31.
Five-limit tuning
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Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths, powers of 5 represent intervals of major thirds. Thus, 5-limit tunings are constructed entirely from extensions of three basic purely-tuned intervals, hence, 5-limit tuning is considered a method for obtaining just intonation. If octaves are ignored, it can be seen as a 2-dimensional lattice of pitch classes extending indefinitely in two directions, however, most tuning systems designed for acoustic instruments restrict the total number of pitches for practical reasons. It is also typical to have the number of pitches in each octave. In that case, the system can also be thought of as an octave-repeating scale of a certain number of pitches per octave. For example, if we have a 5-limit tuning system where the note is C256, then fC =256 Hz. There are several ways to define E above this C. Using thirds, one may go up one factor 5 and down two factors 2, reaching a frequency ratio of 5/4, or using fifths one may go up four factors of 3 and down six factors of 2, reaching 81/64. The prominent notes of a scale are tuned so that their frequencies form ratios of relatively small integers. Here the row headed Natural expresses all these ratios using a common list of natural numbers. In other words, the lowest occurrence of this one-octave scale shape within the series is as a subset of 8 of the 25 harmonics found in the octave from harmonics 24 to 48 inclusive. The three major thirds are correct, and three of the thirds are as expected, but D to F is a semiditone or Pythagorean minor third. As a consequence, we obtain a scale in which EGB and ACE are just minor triads, but the DFA triad doesnt have the minor shape or sound we might expect, being. Furthermore, the BDF triad is not the diminished triad that we would get by stacking two 6,5 minor thirds, being instead, Another way to do it is as follows. The three major thirds are still 5,4, and three of the thirds are still 6,5 with the fourth being 32,27. FAC and CEG still form just major triads, but GBD is now, there are other possibilities such as raising A instead of lowering D, but each adjustment breaks something else. It is evidently not possible to get all seven diatonic triads in the configuration for major, for minor and that demonstrates the need for increasing the numbers of pitches to execute the desired harmonies in tune

32.
7-limit tuning
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7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven, the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example,50,49 is a 7-limit interval, for example, the greater just minor seventh,9,5 Play is a 5-limit ratio, the harmonic seventh has the ratio 7,4 and is thus a septimal interval. Similarly, the chromatic semitone,21,20, is a septimal interval as 21÷7=3. The harmonic seventh is used in the seventh chord and music. Compositions with septimal tunings include La Monte Youngs The Well-Tuned Piano,4, and Lou Harrisons Incidental Music for Corneilles Cinna. The Great Highland Bagpipe is tuned to a ten-note seven-limit scale,1,1,9,8,5,4,4,3,27,20,3,2,5,3,7,4,16,9,9,5. In the 2nd century Ptolemy described the septimal intervals, 7/4, 8/7, 7/6, 12/7, 7/5, and 10/7. Those considering 7 to be dissonant include Gioseffo Zarlino, René Descartes, Jean-Philippe Rameau, Hermann von Helmholtz, A. J. von Öttingen, Hugo Riemann, Colin Brown, the 7-limit tonality diamond, This diamond contains four identities. Similarly, the 2,3,5,7 pitch lattice contains four identities and thus 3-4 axes, laMonte Young created a lattice containing only identities 3 and 7, thus requiring only two axes, for The Well-Tuned Piano. It is possible to approximate 7-limit music using equal temperament, for example 31-ET

33.
Otonality and Utonality
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Otonality and utonality are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone, respectively. An otonality is a collection of pitches which can be expressed in ratios, expressing their relationship to the fixed tone, for example, 1/1, 5/4, and 3/2 form an otonality because they can be written as 4/4, 5/4, 6/4. Every otonality is therefore composed of members of a harmonic series, similarly, the ratios of a utonality share the same numerator. 7/4, 7/5, 7/6, and 1/1 form a utonality, every utonality is therefore composed of members of a subharmonic series. An otonality corresponds to a series of frequencies, or lengths of a vibrating string. Brass instruments naturally produce otonalities, and indeed otonalities are inherent in the harmonics of a fundamental tone. Tuvan Khoomei singers produce otonalities with their vocal tracts, Utonality is the opposite, corresponding to a subharmonic series of frequencies, or an arithmetic series of wavelengths. The arithmetical proportion may be considered as a demonstration of utonality, microtonalists define a just intonation chord as otonal if its odd limit increases on being inverted, utonal if its odd limit decreases, and ambitonal if its odd limit is unchanged. The chord is not inverted in the sense, where C E G becomes E G C or G C E. Instead. A chords odd limit is the largest odd limit of each of the numbers in the chords extended ratio, for example, the major triad 4,5,6 has an odd limit of 5. Its inverse 10,12,15 has an odd limit of 15, Partch said that his 1931 coinage of otonality and utonality was, hastened, by having read Henry Cowells discussion of undertones in New Musical Resources. The 5-limit otonality is simply a just major chord, and the 5-limit utonality is a just minor chord, thus otonality and utonality can be viewed as extensions of major and minor tonality respectively. This chord might be, for example, A♭-C-E♭-G♭ Play, standing alone, it has something of the sound of a dominant seventh, but considerably less dissonant. Utonal chords, while containing the same dyads and roughness as otonal chords, numerary nexus Scale of harmonics Tonality flux Otonality and ADO system at 96-EDO Utonality and EDL system at 96-EDO

34.
Ptolemy's intense diatonic scale
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It is also supported by Giuseppe Tartini. It is produced through a tetrachord consisting of a tone, lesser tone. This is called Ptolemys intense diatonic tetrachord, as opposed to Ptolemys soft diatonic tetrachord, formed by 21/20, 10/9, in comparison to Pythagorean tuning, while both provide just perfect fourths and fifths, the Ptolemaic provides just thirds which are smoother and more easily tuned. Note that D-F is a Pythagorean minor third, D-A is a fifth, F-D is a Pythagorean major sixth. All of these differ from their just counterparts by a syntonic comma and this scale may also be considered as derived from the major chord, and the major chords on top and bottom of it, FAC-CEG-GBD

35.
Pythagorean tuning
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Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio 3,2, which is 702 cents wide. Hence, it is a system of tuning in which the frequency ratios of all intervals are based on the ratio 3,2. This ratio, also known as the perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear. As Novalis put it, The musical proportions seem to me to be particularly correct natural proportions, the so-called Pythagorean tuning was used by musicians up to the beginning of the 16th century. The Pythagorean system would appear to be ideal because of the purity of the fifths, Pythagorean tuning is based on a stack of intervals called perfect fifths, each tuned in the ratio 3,2, the next simplest ratio after 2,1. Since notes differing in frequency by a factor of 2 are given the same name, the purpose of this adjustment is to move the 12 notes within a smaller range of frequency, namely within the interval between the base note D and the D above it. This interval is called the basic octave. For instance, the A is tuned such that its frequency equals 3,2 times the frequency of D—if D is tuned to a frequency of 288 Hz, then A is tuned to 432 Hz. Similarly, the E above A is tuned such that its frequency equals 3,2 times the frequency of A, or 9,4 times the frequency of D—with A at 432 Hz, this puts E at 648 Hz. Since this E is outside the basic octave, it is usual to halve its frequency to move it within the basic octave. Therefore, E is tuned to 324 Hz, a 9,8 above D, the B at 3,2 above that E is tuned to the ratio 27,16 and so on. This frequency is doubled to bring it into the basic octave. When extending this tuning however, a problem arises, no stack of 3,2 intervals will fit exactly into any stack of 2,1 intervals. For instance a stack such as this, obtained by adding one note to the stack shown above A♭–E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯ will be similar. More exactly, it will be about a quarter of a semitone larger, thus, A♭ and G♯, when brought into the basic octave, will not coincide as expected. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a Pythagorean comma, to get around this problem, Pythagorean tuning constructs only twelve notes as above, with eleven fifths between them. For example, one may use only the 12 notes from E♭ to G♯. This, as shown above, the remaining interval is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as one is known as a wolf interval

36.
Scale of harmonics
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The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale, most fret positions appearing on Non-Western string instruments are equal to positions of this scale. Unexpectedly, these positions are actually the corresponding undertones of the overtones from the harmonic series. The distance from the nut to the fret is a number lower than the distance from the fret to the bridge. On the guqin, the end of the dotted scale is a mirror image of the right end. The instrument is played with flageolet tones as well as pressing the strings on the wood, the flageolets appear on the harmonic positions of the overtone series, therefore these positions are marked as the musical scale of this instrument. The flageolet positions also represent the harmonic consonant relation of the string part with the open string. The guqin has one anomaly in its scale, the guqin scale represents the first six harmonics and the eighth harmonic. The seventh harmonic is left out, however this tone is still consonant related to the open string and has a lesser consonant relation to all other harmonic positions. A Vietnamese monochord, called the Đàn bầu, also functions with the scale of harmonics, on this instrument only the right half of the scale is present up to the limit of the first seven overtones. The dots are on the string lengths 1/2, 1/3, 1/4, 1/5, 1/6, partchs tone selection otonality from his utonality and otonality concept are the complement pitches of the overtones. For instance, the frequency ratio 5,4 is equal to 4/5th of the length and 4/5 is the complement of 1/5. Groven used the seljefløyte as basis for his research, the flute uses only the upper harmonic scale. The scale is present on the Moodswinger. Although this functions quite differently than a Guqin, oddly enough the scale occurs on this instrument while it is not played in a just intonation tuning, arithmetic progression Harmonic spectrum Otonality and Utonality Partch, Harry. Genesis Of A Music, An Account Of A Creative Work, Its Roots, article about the overtoning positions and their relation to musical scales

37.
Tonality diamond
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In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality. Equivalently, the diamond may be considered as a set of pitch classes, the tonality diamond is often regarded as comprising the set of consonances of the n-limit. Although originally invented by Max Friedrich Meyer, the tonality diamond is now most associated with Harry Partch, Partch arranged the elements of the tonality diamond in the shape of a rhombus, and subdivided into 2/4 smaller rhombuses. Along the upper side of the rhombus are placed the odd numbers from 1 to n. These intervals are then arranged in ascending order, along the lower left side are placed the corresponding reciprocals,1 to 1/n, also reduced to the octave. These are placed in descending order, at all other locations are placed the product of the diagonally upper- and lower-left intervals, reduced to the octave. This gives all the elements of the tonality diamond, with some repetition, diagonals sloping in one direction form Otonalities and the diagonals in the other direction form Utonalities. One of Partchs instruments, the marimba, is arranged according to the tonality diamond. Harry Partch used the 11-limit tonality diamond, but flipped it 90 degrees, the five- and seven-limit tonality diamonds exhibit a highly regular geometry within the modulatory space, meaning all non-unison elements of the diamond are only one unit from the unison. The five-limit diamond then becomes a regular hexagon surrounding the unison, further examples of lattices of diamonds ranging from the triadic to the ogdoadic diamond have been realised by Erv Wilson where each interval is given its own unique direction. Three properties of the tonality diamond and the contained, All ratios between neighboring ratios are superparticular ratios, those with a difference of 1 between numerator and denominator. Ratios with relatively lower numbers have more space between them than ratios with higher numbers, the system, including the ratios between ratios, is symmetrical within the octave when measured in cents not in ratios. For example, The ratio between 6/5 and 5/4 is 25/24, the ratios with relatively low numbers 4/3 and 3/2 are 203.91 cents apart, while the ratios with relatively high numbers 6/5 and 5/4 are 70.67 cents apart. The ratio between the lowest and 2nd lowest and the highest and 2nd highest ratios are the same, from this we can conclude that the rate of growth of the tonality diamond is asymptotically equal to 2 π2 n 2. The first few values are the important ones, and the fact that the size of the diamond grows as the square of the size of the odd limit tells us that it becomes large fairly quickly. Yuri Landman rewrites Partchs diamond to clarify its relationship to string lengths. Landman flips the ratios and takes the complement string part to them easier to understand. In Partchs ratios, the over number corresponds to the amount of divisions of a vibrating string

38.
Musical temperament
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In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system, tempering is the process of altering the size of an interval by making it narrower or wider than pure. The development of well temperament allowed fixed-pitch instruments to play well in all of the keys. The famous Well-Tempered Clavier by Johann Sebastian Bach takes full advantage of this breakthrough, however, while unpleasant intervals were avoided, the sizes of intervals were still not consistent between keys, and so each key still had its own character. In just intonation, every interval between two pitches corresponds to a whole number ratio between their frequencies, allowing intervals varying from the highest consonance to highly dissonant, for instance,660 Hz /440 Hz constitutes a fifth, and 880 Hz /440 Hz an octave. Such intervals have a stability, or purity to their sound, if, for example, two sound signals with frequencies that vary just by 0. When a musical instrument with harmonic overtones is played, the ear hears a composite waveform that includes a fundamental frequency, the waveform of such a tone is characterized by a shape that is complex compared to a simple waveform, but remains periodic. When two tones depart from exact integer ratios, the shape waveform becomes erratic—a phenomenon that may be described as destabilization, as the composite waveform becomes more erratic, the consonance of the interval also changes. Tempering an interval involves the use of such minor adjustments to enable musical possibilities that are impractical using just intonation. Before Meantone temperament became widely used in the Renaissance, the most commonly used tuning system was Pythagorean tuning, Pythagorean tuning was a system of just intonation that tuned every note in a scale from a progression of pure perfect fifths. This was quite suitable for much of the practice until then. The major third of Pythagorean tuning differed from a just major third by an amount known as syntonic comma, with the correct amount of tempering, the syntonic comma is removed from its major thirds, making them just. This compromise, however, leaves all fifths in this system with a slight beating. Pythagorean tuning also had a problem, which meantone temperament does not solve, which is the problem of modulation. A series of 12 just fifths as in Pythagorean tuning does not return to the pitch, but rather differs by a Pythagorean comma. In meantone temperament, this effect is more pronounced. The use of 53 equal temperament provides a solution for the Pythagorean tuning, when building an instrument, this can be very impractical. Well temperament is the given to a variety of different systems of temperament that were employed to solve this problem, in which some keys are more in tune than others

39.
Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step, the standard pitch has not always been 440 but has varied and generally risen over the past few hundred years. For example, some music has been written in 19-TET and 31-TET, in Western countries, when people use the term equal temperament without qualification, they usually mean 12-TET. To avoid ambiguity between equal temperaments that divide the octave and ones that divide some other interval, the equal division of the octave. According to this system, 12-TET is called 12-EDO, 31-TET is called 31-EDO. Other instruments, such as wind, keyboard, and fretted instruments, often only approximate equal temperament. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones, use tuning similar to string ensembles, the two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu in 1584 and Simon Stevin in 1585. Kenneth Robinson attributes the invention of equal temperament to Zhu Zaiyu, Zhu Zaiyu is quoted as saying that, in a text dating from 1584, I have founded a new system. I establish one foot as the number from which the others are to be extracted, altogether one has to find the exact figures for the pitch-pipers in twelve operations. Kuttner disagrees and remarks that his claim cannot be considered correct without major qualifications, kuttner proposes that neither Zhu Zaiyu or Simon Stevin achieved equal temperament, and that neither of the two should be treated as inventors. The origin of the Chinese pentatonic scale is traditionally ascribed to the mythical Ling Lun, allegedly his writings discussed the equal division of the scale in the 27th century BC. However, evidence of the origins of writing in this period in China is limited to rudimentary inscriptions on oracle bones, an approximation for equal temperament was described by He Chengtian, a mathematician of Southern and Northern Dynasties around 400 AD. He came out with the earliest recorded approximate numerical sequence in relation to equal temperament in history,900849802758715677638601570536509.5479450, historically, there was a seven-equal temperament or hepta-equal temperament practice in Chinese tradition. Zhu Zaiyu, a prince of the Ming court, spent thirty years on research based on the equal temperament idea originally postulated by his father and he described his new pitch theory in his Fusion of Music and Calendar 乐律融通 published in 1580. An extended account is given by Joseph Needham. Similarly, after 84 divisions the length was divided by a factor of 128,84 =27 =128, according to Gene Cho, Zhu Zaiyu was the first person to solve the equal temperament problem mathematically. Matteo Ricci, a Jesuit in China recorded this work in his personal journal, in 1620, Zhus work was referenced by a European mathematician

40.
Whole tone scale
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In music, a whole tone scale is a scale in which each note is separated from its neighbours by the interval of a whole tone. This effect is especially emphasised by the fact that triads built on such scale tones are augmented, indeed, one can play all six tones of a whole tone scale simply with two augmented triads whose roots are a major second apart. Since they are symmetrical, whole tone scales do not give an impression of the tonic or tonality. The composer Olivier Messiaen called the tone scale his first mode of limited transposition. The composer and music theorist George Perle calls the whole tone scale interval cycle 2, since there are only two possible whole tone scale positions, it is either C20 or C21. For this reason, the tone scale is also maximally even. Due to this symmetry, the hexachord consisting of the scale is not distinct under inversion or more than one transposition. Use of the whole tone scale can be traced at least as far back as Johann Sebastian Bach. The concluding chorale movement of his cantata O Ewigkeit du Donnerwort BWV60, opens with four notes from the tone scale, Mozart also used the scale in his Musical Joke, for strings. Further examples can be found in the works of Rimsky-Korsakov, the sea kings music in Sadko and also in Scheherezade, colles names as the childhood of the whole-tone scale the music of Berlioz and Schubert in France and then Russians Glinka and Dargomyzhsky. The sense of mystery and ambiguity here even extends to the title of the piece, though the whole-tone scale is prominent in much of his music after 1905 when he encountered Debussy, it serves simply to fit the motifs over augmented chords. The same motifs return from the whole-tone to the scale without emphasizing the contrast. The first of Alban Bergs Seven Early Songs opens with a whole-tone passage both in the accompaniment and in the vocal line that enters a bar later. Berg also quotes the Bach chorale setting referred to above in his Violin Concerto. The last four notes of the 12-tone row Berg used are B, C♯, E♭ and F, an early instance of the use of the scale in jazz writing can be found in Don Redman’s “Chant of the Weed”. In 1958, Gil Evans recorded an arrangement that gives striking coloration to passages featuring whole-tone harmonies, however, these are only the most overt examples of the use of this scale in jazz. Art Tatum and Thelonious Monk are two pianists who used the tone scale extensively and creatively. Thelonious Monks Four in One and Trinkle-Tinkle are fine examples of this, a prominent example of the whole tone scale that made its way into pop music are bars 3 and 4 of the opening of Stevie Wonders 1972 song You are the Sunshine of my Life

41.
15 equal temperament
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In music,15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is a tempered scale derived by dividing the octave into 15 equal steps. Each step represents a ratio of 15√2, or 80 cents. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of 5 equal divisions of the octave, each of which resembles the Slendro scale in Indonesian gamelan,15 equal temperament is not a meantone system. Guitars have been constructed for 15-ET tuning, the American musician Wendy Carlos used 15-ET as one of two scales in the track Afterlife from the album Tales of Heaven and Hell. Easley Blackwood, Jr. has written and recorded a suite for 15-ET guitar, Blackwood believes that 15 equal temperament, is likely to bring about a considerable enrichment of both classical and popular repertoire in a variety of styles. Here are the sizes of some common intervals in 15-ET, 15-ET matches the 7th and 11th harmonics well, but only matches the 3rd and 5th harmonics roughly. The perfect fifth is more out of tune than in 12-ET, 19-ET, or 22-ET, and the third in 15-ET is the same as the major third in 12-ET. Unlike 12-ET and 19-ET, 15-ET matches the 11,8 and 16,11 ratios, 15-ET also has a neutral second and septimal whole tone. To construct a third, one must stack two intervals of different sizes, whereas one can divide both the minor third and perfect fourth into two equal intervals. Ivor Darreg, 15-TONE SCALE SYSTEM, Sonic-Arts. org, brewt, Fifteen note equal temperament tutorial

42.
17 equal temperament
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In music,17 tone equal temperament is the tempered scale derived by dividing the octave into 17 equal steps. Each step represents a ratio of 17√2, or 70.6 cents. Alexander J. Ellis refers to a tuning of seventeen tones based on fourths and fifths as the Arabic scale. This 17-tone system remained the primary theoretical system until the development of the tone scale. 17-ET is the tuning of the Regular diatonic tuning in which the tempered fifth is equal to 705.88 cents. 17-ET is where every other step in the 34-ET scale is included, conversely 34-ET is a subdivision of 17-ET. The 17-tone Puzzle — And the, microtonalismo Heptadecatonic System Applications Georg Hajdus 1992 ICMC paper on the 17-tone piano project ProyectoXVII Heptadecatonic System Applications project XVII - Peruvian

43.
19 equal temperament
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In music,19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a ratio of 19√2, or 63.16 cents. 19-edo is the tuning of the temperament in which the tempered perfect fifth is equal to 694.737 cents. Division of the octave into 19 equal-width steps arose naturally out of Renaissance music theory, the greater diesis, the ratio of four minor thirds to an octave was almost exactly a nineteenth of an octave. Interest in such a system goes back to the 16th century. Costeley understood and desired the circulating aspect of this tuning, in 1577, music theorist Francisco de Salinas in effect proposed it. Salinas discussed 1⁄3-comma meantone, in which the fifth is of size 694.786 cents, the fifth of 19-edo is 694.737, which is less than a twentieth of a cent narrower, imperceptible and less than tuning error. Salinas suggested tuning nineteen tones to the octave to this tuning, in the 19th century, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone temperaments he regarded as better, such as 50-edo. The composer Joel Mandelbaum wrote his Ph. D. thesis on the properties of the 19-edo tuning, Mandelbaum and Joseph Yasser have written music with 19-edo. Easley Blackwood has stated that 19-edo makes possible a substantial enrichment of the tonal repertoire, here are the sizes of some common intervals and comparison with the ratios arising in the harmonic series, the difference column measures in cents the distance from an exact fit to these ratios. For reference, the difference from the fifth in the widely used 12 equal temperament is 1.955 cents. Beta scale Elaine Walker Levy, Kenneth J. Costeleys Chromatic Chanson, Annales Musicologues, Moyen-Age et Renaissance, Tome III, howe, Hubert S. Jr. 19-Tone Theory and Applications, Aaron Copland School of Music at Queens College. Sethares, William A. Tunings for 19 Tone Equal Tempered Guitar, Experimental Musical Instruments, Vol. VI, hair, Bailey, Morrison, Pearson and Parncutt, Rehearsing Microtonal Music, Grappling with Performance and Intonational Problems, Microtonalism. ZiaSpace. com - 19tet downloadable mp3s by Elaine Walker of Zia, the Music of Jeff Harrington, Parnasse. com. Jeff Harrington is a composer who has several pieces for piano in the 19-TET tuning. Chris Vaisvil, GR-20 Hexaphonic 19-ET Guitar Improvisation Arto Juhani Heino, Artone 19 Guitar Design, naming the 19 note scale Parvatic

Harmonic series (music)
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A harmonic series is the sequence of sounds where the base frequency of each sound is an integer multiple of the lowest base frequency. Pitched musical instruments are based on an approximate harmonic oscillator such as a string or a column of air. At these resonant frequencies, waves travel in both directions along the string or air column, reinfo

1.
Harmonic series of a string with terms written as reciprocals (2/1 written as 1/2)

Musical scale
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In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is a scale. Some scales contain different pitches when ascending than when descending, for example, the Melodic minor scale. Due to the principle of equivalence, scales are generally considered to span a single octave.

1.
Ascending and descending chromatic scale Play (help · info)

Just intonation
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In music, just intonation or pure intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a pure or just interval, pure intervals are important in music because they correspond to the vibrational patterns found in physical objects which correlate to hu

1.
Harmonic series, partials 1–5 numbered Play (help · info).

Limit (music)
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In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony, roughly speaking, the larger the limit number, the more harmon

1.
Overtone series, partials 1-5 numbered Play (help · info).

Harmonic
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A harmonic is any member of the harmonic series, the divergent infinite series. Every term of the series after the first is the mean of the neighboring terms. The phrase harmonic mean likewise derives from music, the term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and ot

1.
Table of harmonics of a stringed instrument with colored dots indicating which positions can be lightly fingered to generate just intervals up to the 7th harmonic

2.
The nodes of a vibrating string are harmonics.

Modulation (music)
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In music, modulation is most commonly the act or process of changing from one key to another. This may or may not be accompanied by a change in key signature, modulations articulate or create the structure or form of many pieces, as well as add interest. Treatment of a chord as the tonic for less than a phrase is considered tonicization, Modulation

1.
Example of modulation from the tonic to the dominant as in vocal music. Play (help · info)

Synthesizer
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A synthesizer is an electronic musical instrument that generates electric signals that are converted to sound through instrument amplifiers and loudspeakers or headphones. Synthesizers may either imitate instruments like piano, Hammond organ, flute, vocals, natural sounds like ocean waves, etc. or generate new electronic timbres. Synthesizers witho

Transposition (music)
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In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval. For example, one might transpose a piece of music into another key. Similarly, one might transpose a tone row or a collection of pitches such as a chord so that it begins on another pitch. The transposition of a se

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Transposition example from Koch Play top (help · info) Play bottom (help · info). The melody on the first line is in the key of D, while the melody on the second line is identical except that it is major third lower, in the key of B ♭.

Fundamental frequency
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The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the frequency is the lowest frequency sinusoidal in the sum. In some contexts,

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Vibration and standing waves in a string, The fundamental and the first six overtones

Beauty in the Beast
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The album uses alternate musical tunings and scales, influenced by jazz and world music. On the back she includes a quote by Van Gogh, I am always doing what I cannot do yet, as the liner notes state, the entire album is synthesized, meaning that All the music and sounds heard on this recording were directly digitally generated. This eliminates all

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Beauty in the Beast

Square number
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In mathematics, a square number or perfect square is an integer that is the square of an integer, in other words, it is the product of some integer with itself. For example,9 is a number, since it can be written as 3 × 3. The usual notation for the square of a n is not the product n × n. The name square number comes from the name of the shape, anot

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m = 1 2 = 1

Triangular number
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A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangular number is the number of dots composing a triangle with n dots on a side and it represents the number of distinct pairs that can be selected from n +1 objects, and it is read aloud as n plus one choose t

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The first six triangular numbers

Note (music)
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In music, the term note has three primary meanings, A sign used in musical notation to represent the relative duration and pitch of a sound, A pitched sound itself. Notes are the blocks of much written music, discretizations of musical phenomena that facilitate performance, comprehension. In the former case, one note to refer to a specific musical

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Names of some notes without accidentals

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The note A or La

Octave
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In music, an octave or perfect octave is the interval between one musical pitch and another with half or double its frequency. It is defined by ANSI as the unit of level when the base of the logarithm is two. The octave relationship is a phenomenon that has been referred to as the basic miracle of music. The most important musical scales are writte

International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning

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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code

JSTOR
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JSTOR is a digital library founded in 1995. Originally containing digitized back issues of journals, it now also includes books and primary sources. It provides full-text searches of almost 2,000 journals, more than 8,000 institutions in more than 160 countries have access to JSTOR, most access is by subscription, but some older public domain conte

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The JSTOR front page

Musical tuning
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In music, there are two common meanings for tuning, Tuning practice, the act of tuning an instrument or voice. Tuning systems, the systems of pitches used to tune an instrument. Tuning is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones, Tuning is usually based on a

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Two differently tuned thirds: Just major third Play (help · info).

Pitch (music)
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Pitch can be determined only in sounds that have a frequency that is clear and stable enough to distinguish from noise. Pitch is a major attribute of musical tones, along with duration, loudness. Pitch may be quantified as a frequency, but pitch is not a purely objective physical property, Pitch is an auditory sensation in which a listener assigns

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In musical notation, the different vertical positions of notes indicate different pitches. Play top (help · info) & Play bottom (help · info)

Cent (music)
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The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each, alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquets suggestion. It has beco

Savart
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The savart /səˈvɑːr/ is a unit of measurement for musical pitch intervals. One savart is equal to one thousandth of a decade,3.9863 cents, musically, in just intonation, the interval of a decade is precisely a just major twenty-fourth, or, in other words, three octaves and a just major third. Today the savart has largely replaced by the cent and th

Interval (music)
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In music theory, an interval is the difference between two pitches. In Western music, intervals are most commonly differences between notes of a diatonic scale, the smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones and they can be formed using the notes of various kinds of non-diatonic scales. Some o

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Melodic and harmonic intervals. Play (help · info)

Interval ratio
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In music, an interval ratio is a ratio of the frequencies of the pitches in a musical interval. For example, a just perfect fifth is 3,2,1.5, if the A above middle C is 440 Hz, the perfect fifth above it would be E, at 660 Hz, while the equal tempered E5 is 659.255 Hz. Ratios have a relationship to string length, for example stopping a string at tw

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3-limit 9:8 major tone Play (help · info).

Pitch class
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In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e. g. the pitch class C consists of the Cs in all octaves. The pitch class C stands for all possible Cs, in whatever octave position, important to musical set theory, a pitch class is, all pitches related to each other by octave, enharmonic equivalence, or bot

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All Cs possible on a piano (except C 8, available on grand) Play (help · info).

Consonance and dissonance
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In music, consonance and dissonance form a structural dichotomy in which the terms define each other by mutual exclusion, a consonance is what is not dissonant, and reciprocally. However, a finer consideration shows that the forms a gradation. Consonance is associated with sweetness, pleasantness and acceptability and dissonance with harshness, unp

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The perfect octave, a consonant interval Play (help · info)

List of pitch intervals
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Below is a list of intervals exprimable in terms of a prime limit, completed by a choice of intervals in various equal subdivisions of the octave or of other intervals. For instance, the limit of the just perfect fourth is 3, there exists another type of limit, the odd limit, a concept used by Harry Partch, but it is not used here. The term limit w

List of intervals in 5-limit just intonation
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The intervals of 5-limit just intonation are ratios involving only the powers of 2,3, and 5. The fundamental intervals are the superparticular ratios 2/1, 3/2 and 5/4 and that is, the notes of the major triad are in the ratio 1, 5/4, 3/2 or 4,5,6. In all tunings, the third is equivalent to two major seconds. However, because just intonation does no

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Pythagorean minor third

Microtonal music
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Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, which are also called microintervals. It may also be extended to any music using intervals not found in the customary Western tuning of twelve equal intervals per octave. Microtonal music can refer to any music containing microtones, therefore it

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Composer Charles Ives chose the chord above as a good candidate for a "fundamental" chord in the quarter tone scale, akin not to the tonic but to the major chord of traditional tonality (Boatwright 1971, 8–9). Play (help · info) or play (help · info)

Harry Partch's 43-tone scale
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The 43-tone scale is a just intonation scale with 43 pitches in each octave, invented and used by Harry Partch. Almost all of Partchs music is written in the 43-tone scale, Partch chose the 11 limit as the basis of his music, because the 11th harmonic is the first that is utterly foreign to Western ears. Here are all the ratios within the octave wi

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The Quadrangularis Reversum, one of Partch's instruments featuring the 43-tone scale

Hexany
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In music theory, the hexany is a six-note just intonation structure, with the notes placed on the vertices of an octahedron, equivalently the faces of a cube. The notes are arranged so that edge of the octahedron joins together notes that make a consonant dyad. This makes a musical geometry with the form of the octahedron. The edges of the octahedr

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Regular Octahedron.

Five-limit tuning
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Powers of 2 represent intervallic movements by octaves. Powers of 3 represent movements by intervals of perfect fifths, powers of 5 represent intervals of major thirds. Thus, 5-limit tunings are constructed entirely from extensions of three basic purely-tuned intervals, hence, 5-limit tuning is considered a method for obtaining just intonation. If

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Primary triads in C Play (help · info).

7-limit tuning
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7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven, the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example,50,49 is a 7-limit interval, for example, the greater just minor seventh,9,5 Play is a 5-limit ratio, the harmonic seventh has the ratio 7,4 and is

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Harmonic seventh Play (help · info), septimal seventh.

Otonality and Utonality
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Otonality and utonality are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone, respectively. An otonality is a collection of pitches which can be expressed in ratios, expressing their relationship to the fixed tone, for example, 1/1, 5/4, and 3/2 form an otonality because

Ptolemy's intense diatonic scale
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It is also supported by Giuseppe Tartini. It is produced through a tetrachord consisting of a tone, lesser tone. This is called Ptolemys intense diatonic tetrachord, as opposed to Ptolemys soft diatonic tetrachord, formed by 21/20, 10/9, in comparison to Pythagorean tuning, while both provide just perfect fourths and fifths, the Ptolemaic provides

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Diatonic scale on C, equal tempered Play (help · info) and Ptolemy's intense or just Play (help · info).

Pythagorean tuning
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Pythagorean tuning is a tuning of the syntonic temperament in which the generator is the ratio 3,2, which is 702 cents wide. Hence, it is a system of tuning in which the frequency ratios of all intervals are based on the ratio 3,2. This ratio, also known as the perfect fifth, is chosen because it is one of the most consonant and easiest to tune by

Scale of harmonics
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The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string. This musical scale is present on the guqin, regarded as one of the first string instruments with a musical scale, most fret positions appearing on Non-Western string instruments are equal to positions of this scale. Unexpectedly, th

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The Guqin

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Scale of harmonics on the Moodswinger

Tonality diamond
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In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality. Equivalently, the diamond may be considered as a set of pitch classes, the tonality diamond is often regarded as comprising the set of consonances of the n-limit. Although originally invented by Max Fr

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The Quadrangularis Reversum, an instrument constructed by Harry Partch based on the 11-limit tonality diamond

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Interlocked movement from a major to a minor triad. Play (help · info)

Musical temperament
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In musical tuning, a temperament is a tuning system that slightly compromises the pure intervals of just intonation to meet other requirements. Most modern Western musical instruments are tuned in the equal temperament system, tempering is the process of altering the size of an interval by making it narrower or wider than pure. The development of w

Equal temperament
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An equal temperament is a musical temperament, or a system of tuning, in which the frequency interval between every pair of adjacent notes has the same ratio. In equal temperament tunings, the interval is often found by dividing some larger desired interval, often the octave. That resulting smallest interval, 1⁄12 the width of an octave, is called

Whole tone scale
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In music, a whole tone scale is a scale in which each note is separated from its neighbours by the interval of a whole tone. This effect is especially emphasised by the fact that triads built on such scale tones are augmented, indeed, one can play all six tones of a whole tone scale simply with two augmented triads whose roots are a major second ap

15 equal temperament
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In music,15 equal temperament, called 15-TET, 15-EDO, or 15-ET, is a tempered scale derived by dividing the octave into 15 equal steps. Each step represents a ratio of 15√2, or 80 cents. Because 15 factors into 3 times 5, it can be seen as being made up of three scales of 5 equal divisions of the octave, each of which resembles the Slendro scale in

17 equal temperament
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In music,17 tone equal temperament is the tempered scale derived by dividing the octave into 17 equal steps. Each step represents a ratio of 17√2, or 70.6 cents. Alexander J. Ellis refers to a tuning of seventeen tones based on fourths and fifths as the Arabic scale. This 17-tone system remained the primary theoretical system until the development

19 equal temperament
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In music,19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps. Each step represents a ratio of 19√2, or 63.16 cents. 19-edo is the tuning of the temperament in which the tempered perfect fifth is equal to 694.737 cents. Division of the octave into 19 equal-width steps arose

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Approximate size in cents of the 144 intervals in D-based quarter-comma meantone tuning. Interval names are given in their standard shortened form. Pure intervals are shown in bold font. Wolf intervals are highlighted in red.